OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 24 — Nov. 23, 2009
  • pp: 21442–21451

Optics InfoBase > Optics Express > Volume 17 > Issue 24 > Page 21442

« Show journal navigation

Time delays in the synchronization of chaotic coupled lasers with feedback

Jhon F. Martinez Avila and J. R. Rios Leite  »View Author Affiliations


Optics Express, Vol. 17, Issue 24, pp. 21442-21451 (2009)
http://dx.doi.org/10.1364/OE.17.021442


View Full Text Article

Acrobat PDF (244 KB)


Advanced Search

Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Isochrony and time leadership was studied in the synchronized excitable behavior of coupled chaotic diode lasers. Each unit of the system had chaos due to feedback with a fixed delay time. The inter-units coupling signal had a second, independent, characteristic time. Synchronized excitable spikes present isochronous, time leading or time lagging behavior whose stability is shown to depend on a simple relation between the feedback and the coupling times. Experiments on the synchronized low frequency fluctuations of two optically coupled semiconductor lasers and numerical calculations with coupled laser equations verify the predicted stability conditions for synchronization. Synchronism with intermittent time leadership exchange was also observed and characterized.

© 2009 Optical Society of America

1. Introduction

Many dynamical systems in nature have chaos due to feedback. A characteristic time, τF, associated to the feedback is therefore embedded in the system response. As two or more of such systems are coupled, another independent time, τC, corresponding to the time of flight of the coupling signal, enters in the dynamical description of the global system. We present here how the relation between these two times determines the possible time delays in chaos synchronization. New features of time leadership competition in synchronized chaos between coupled pairs of systems with feedback are found when feedback and inter-coupling times have the same order of magnitude. Experimental and numerically, the stability of isochronous chaos synchronism for identical systems occurs only for specific relations between these times. We also show that time delayed and time advanced synchronism as well as synchronism with intermittent leadership exchange are also quantitatively determined by the ratio of these times. Our case is made with pairs of semiconductor diode lasers. However, the properties of synchronization in complex systems extends far beyond physical devices [1

A. Pikovsky, M. Rosenblum, and J. Kurths, “Synchronization: A Universal Concept in Nonlinear Sciences,” (Cambridge Nonlinear Science Series, 2003)

], reaching the subject of neural sciences [2

R. Vicente, L. L Gollo, C. R. Mirasso, I. Fisher, and Gordon Pipa, “Dynamical relaying can yield zero time lag neuronal synchrony despite long conduction delays,” PNAS- Proceedings of the National Academy of Sciences of the USA 105, 17157–17162 ( 2008). [CrossRef]

].

The study of synchronism with chaotic lasers spreads for more than a decade [3

T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Y. Shimizu, “Observation of Synchronization in Laser Chaos,” Phys. Rev. Lett. 72, 3502–3505 ( 1994). [CrossRef] [PubMed]

, 4

R. Roy and K. S. Thornburg, “Experimental Synchronization of Chaotic Lasers,” Phys. Rev. Lett. 72, 2009–2012 ( 1994). [CrossRef] [PubMed]

, 5

Y. Liu, P.C. de Oliveira, M.B. Danailov, and J. R. Rios Leite, “Chaotic and periodic passive Q switching in coupled CO2 lasers with a saturable absorber,” Phys. Rev. A 50, 3464–3470 ( 1994). [CrossRef] [PubMed]

, 6

R. Meucci, F. Salvadori, M. V. Ivanchenko, K. Al Naimee, C. Zhou, F. T. Arecchi, S. Boccaletti, and J. Kurths, “Synchronization of spontaneous bursting in a CO2 laser,” Phys. Rev. E 74, 066207-1 0662070-6 ( 2006). [CrossRef]

, 10

I. Oowada, H. Ariizumi, M. Li, S. Yoshimori, A. Uchida, K. Yoshimura, and P. Davis, “Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback,” Opt. Express 17, 10025–10034 ( 2009). [CrossRef] [PubMed]

]. Of relevant interest for applications are the results on the synchronization of semiconductor lasers [11

Y. Takiguchi, H. Fujino, and J. Ohtsubo, “Experimental synchronization of chaotic oscillations in externally injected semiconductor lasers in a low-frequency fluctuations regime,” Opt. Lett. 24, 1570–1572 ( 1999). [CrossRef]

, 12

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 ( 2005). [CrossRef] [PubMed]

] for encrypted communication. With optical feedback diode lasers can present chaos in the form of very fast output power fluctuations, at a time scale of picoseconds. Superimposed on these, irregular power drops occur in amuch slower time scale (order of hundreds of nanoseconds and longer) corresponding to the so called low frequency fluctuations (LFF) [13

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quant. Semiclass. Opt. 7, 87–143 ( 1995). [CrossRef]

]. The LFF drops in single lasers are a current subject of studies and have been associated to spikes of excitable systems [14

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E 55, 6414–6418 ( 1997). [CrossRef]

, 15

J. F. Martinez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, “Experimental Deterministic Coherence Resonance,”Phys. Rev. Lett. 93, 144101-1–144101-4 ( 2004). [CrossRef]

, 16

J. F. Martinez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, “Coarse Grained Variables and Deterministic Chaos in an Excitable System,” Phys. Rev. Lett. 100, 044101-1–044101-4 ( 2008). [CrossRef]

]. Coupled diode lasers show time advanced and time lagging synchronization via unidirectional coupling in master-slave configuration [17

C. Masoller, “Anticipation in the Synchronization of Semiconductor Lasers with Optical Feedback,” Phys. Rev. Lett. 86, 2782–2785 ( 2001). [CrossRef] [PubMed]

, 18

P. Rees, P. S. Spencer, I. Pierce, S. Sivaprakasam, and K. A. Shore “Anticipated chaos in a non symmetric coupled external-cavity-laser system” Source: Phys. Rev. A 68, 033818-1–033818-7 ( 2003).

] and in mutually coupled systems [19

T. Heil, I. Fischer, W. Elssser, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86, 795–798 ( 2001). [CrossRef] [PubMed]

, 18

P. Rees, P. S. Spencer, I. Pierce, S. Sivaprakasam, and K. A. Shore “Anticipated chaos in a non symmetric coupled external-cavity-laser system” Source: Phys. Rev. A 68, 033818-1–033818-7 ( 2003).

, 20

J. F. M. Avila, R. Vicente, J. R. Rios Leite, and C. R. Mirasso “Synchronization properties of bidirectionally coupled semiconductor lasers under asymmetric operating conditions”, Phys. Rev. E 75, 066202-1–066202-6 ( 2007). [CrossRef]

]. Symmetrically coupled pairs of lasers, without feedback, were shown to have unstable isochronous chaotic pulsation [19

T. Heil, I. Fischer, W. Elssser, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86, 795–798 ( 2001). [CrossRef] [PubMed]

, 21

J. Mulet, C. R. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B: Quant. Semiclass. Opt. 6, 97–105 ( 2004). [CrossRef]

, 22

E. A. Rogers-Dakin, J. García-Ojalvo, D. J. DeShazer, and R. Roy, “Synchronization and symmetry breaking in mutually coupled fiber lasers,” Phys. Rev. E 73, 045201 ( 2006). [CrossRef]

]. In the experiments and calculations with coupled lasers without feedback, the time leading lasers always appears with its power drops displaced by one unit of the coupling time, τC . The use of intermediate relaying system was demonstrated to give stable isochrony [23

I. Fischer, R. Vicente, J. M. Buldú, M. Peil, and C. R. Mirasso, “Zero-Lag Long-Range Synchronization via Dynamical Relaying,” Phys. Rev. Lett. 97, 123902-1–123902-4 ( 2006). [CrossRef]

, 24

I. B. Schwartz and L. B. Shaw, “Isochronal synchronization of delay-coupled systems,” Phys. Rev. E 75, 046207-1–046207-5 ( 2007). [CrossRef]

]. Isochronous synchronization has also been investigated [24

I. B. Schwartz and L. B. Shaw, “Isochronal synchronization of delay-coupled systems,” Phys. Rev. E 75, 046207-1–046207-5 ( 2007). [CrossRef]

, 25

E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E 73, 066214-1–066214-4 ( 2006). [CrossRef]

, 26

N. Gross, W. Kinzel, I. Kanter, M. Rosenbluh, and L. Khaykovich, “Synchronization of mutually versus unidirectionally coupled chaotic semiconductor lasers,” Opt. Commun. 264, 464–468 ( 2006). [CrossRef]

] for lasers with feedback with the studies focused on the fast laser fluctuations. Differently, herein we study the synchronization of the low frequency fluctuations (LFF), known to appear at the scale of hundreds of nanoseconds and slower. Thus, in our case the dynamics in the coupled systems is much faster than the synchronized events which have the properties of excitable spikes. So, our results refer to synchronism of excitable dynamical systems.

2. Experimental setup

The schema of the experiments is given in figure 1. Optical feedback was created in each laser by a retro-reflecting external mirror. Their feedback return times, τ F1 and τ F2, were set equal to within 1% precision (both named from hereon τF ). The time of travel of the coupling signal between the lasers, τC , was independently controlled with respect to τF . Small changes in either of the times, on the scale of fraction of nanoseconds do not alter the properties of the synchronism. Thus the results, like the LFF phenomenon in single laser with feedback, are robust with respect to optical phase changes. A consequence of such behavior is that our observations are consistent with the synchronization of excitable systems [27

M. Cisak, F. Marino, R. Toral, and S. Balle, “Mechanism of Anticipating Synchronization in Excitable Systems,” Phys. Rev. Lett. 93, 114102-1–114102-4 ( 2004).

]. These authors show how unidirectionally coupled excitable systems can presents one of the systems always with a lower threshold for excitability. Their coupled systems synchronize to an external common signal. In our case the coupled systems have fast fluctuations and so, no external source of excitation is necessary.

Fig. 1. Experimental setup for the synchronization of Low Frequency Fluctuations in optically coupled laser.

The experiments were done with two SDL 5401 GaAlAs semiconductor lasers, named here as Laser 1 and Laser 2, both with solitary threshold current of 21 mA and emitting at 805 nm. They were thermally stabilized to 0.01 K and could be temperature tuned to have their optical frequencies within 2 Ghz separation. The feedback time of the lasers was τF =10ns and the strength were measured by the threshold reduction parameter, ξ which is the percentual variation of the threshold pump current as we consider the laser with and without feedback [13

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quant. Semiclass. Opt. 7, 87–143 ( 1995). [CrossRef]

]. Symmetrical optical coupling between the lasers was produced with 50% beam splitters as shown in figure 1. The time of flight of the light between the lasers was varied between 5 and 20 ns. The threshold reductions due to cross input power were used to quantify the coupling strengths whose contribution to our studies will be detailed elsewhere. Manipulating the laser currents and temperatures, LFF synchronism was obtained where to each power drop in laser 1 corresponds a drop in laser 2 and vice versa.

3. Results

Typical experimental segments of the power output of the two lasers, with three events of the synchronized low frequency fluctuations (LFF) drops, are shown in figure 2. Each laser intensity was detected by a 1.5 GHz bandwidth photodiode. Simultaneous data series were acquired with a two channel digital oscilloscope having a bandwidth of 1 GHz and a maximum sampling rate of 5 GSamples/second. One of the lasers could always be the time leader by setting its pump current or its optical frequency [19

T. Heil, I. Fischer, W. Elssser, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86, 795–798 ( 2001). [CrossRef] [PubMed]

] higher. However, as careful tuning of the laser currents and optical frequencies is made, within the same long LFF synchronized time series the LFF drops of laser 1 can appear isochronous, leading or lagging these of laser 2. Such were the conditions of figure 2 where the time leadership is interchanged between drops one and three, while the second LFF drops are isochronous. Using long pairs of data series, coarse grained to a 1ns time resolution [16

J. F. Martinez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, “Coarse Grained Variables and Deterministic Chaos in an Excitable System,” Phys. Rev. Lett. 100, 044101-1–044101-4 ( 2008). [CrossRef]

], we measured ΔTi =T 1i -T 2i , the delay between the LFF drop i of laser 1 and laser 2.

Fig. 2. Segment of experimental time series of the power of (a) laser 1 and (b) laser 2, showing the low frequency synchronism and including the interchange of time delay between the lasers. τF =10 ns and τC =10 ns.

The main result of this work is to show that, within any LFF synchronized evolution, for any pair of excited drops the allowed delays are given by

Δ Ti= mi∙ τC− ni∙ τF
(1)

where mi ≠0 and ni are small integers. Here we only give indications of mi =±1 but preliminary results with very long numerical series show rare events with mi >1. Equation 1 covers various previously studied cases in the literature and confirms novel observations. For instance, with the lasers having no feedback, τF =0, it verified that isochrony is not allowed [19

T. Heil, I. Fischer, W. Elssser, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86, 795–798 ( 2001). [CrossRef] [PubMed]

]. Other cases with feedback but forbidding isochrony are given below. The equation determines ΔTi when one laser is always leading or lagging which happens for a single pair (mi ,ni ) through the whole time series. Furthermore, it is also valid for cases with varying values of (mi ,ni ) within the same series, obtained from symmetrical and nearly symmetrical systems. The LFF synchronism then appears with intermittent switching of the time delay that can interchange the leading subsystem. The evidence of equation 1 as a property of the LFF synchronism was obtained inspecting many different feedback and coupling delay times in experimental and numerical series. The origin of equation 1 is present in the intensity cross-correlation functions of the fast fluctuations of coupled lasers. These correlations, obtained with sub-nanosecond resolution, show recurrent narrow spikes (width of hundred of picoseconds) at positions and intervals given by Eq. 1.

Let us proceed presenting our experiments along with the numerical-theoretical results extracted from a system of differential equations that describe two mono-mode lasers with optical feedback and optical coupling. The model correspond to a set of modified equations for the lasers with feedback [28

R. Lang and K. Kobayashi, “External Optical Feedback Effects on Semiconductor Injection Laser Properties,” IEEE J. Quantum Electron. QE-16, 347–355 ( 1980). [CrossRef]

], including symmetrical optical coupling and assumed to have optical frequence detuning Ω=ω 2-ω 1:

dEi (t)dt= ( 1+iα)2 [G ( Ni)− 1 τp] Ei (t)+κ Ei ( t− τF)exp ( iϕi)

+ γE j ≠i ( t− τC)exp (−i Ωj) ( t− τC)
(2)
dNi (t)dt= Ji− Ni (t) τs−G ( Ni) ∣ Ei (t)∣2,
(3)

where i,j=1,2, Ω1=+Ω, Ω2=-Ω and each laser gain is given by

G ( Ni)= G0 ( Ni− N0) 1+ε ∣ Ei (t)∣2
(4)

The definition of the various parameters and their typical values are well discussed in the literature [13

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quant. Semiclass. Opt. 7, 87–143 ( 1995). [CrossRef]

, 15

J. F. Martinez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, “Experimental Deterministic Coherence Resonance,”Phys. Rev. Lett. 93, 144101-1–144101-4 ( 2004). [CrossRef]

]. For each laser, Ej is the radiation field amplitude, ωi is the optical frequency, α is the factor describing amplitude to phase conversion, G is the amplifying gain, Nj (t) the carriers inversion population, τp the photon lifetime of the internal laser cavity, τs the carriers lifetime, Jj the threshold normalized pump currents and N 0 the inversion population for medium transparency. Each feedback field has an amplitude coefficient κ and feedback time τF . The optical couplings are linearly additive E field with coefficient γ and time delay τC for the field of one laser to reach the other one. Physical causality demands that both τF and τC be positive. The fixed phases of each laser are ϕi . Their contribution to the LFF events and the lasers synchronizations is not significant [7

J. M. Buldú, T. Heil, I. Fischer, M. C. Torrent, and J. Garcia-Ojalvo, “Episodic Synchronization via Dynamic Injection,” Phys. Rev. Lett. 96, 024102-1–024102-4 ( 2006) [CrossRef]

, 10

I. Oowada, H. Ariizumi, M. Li, S. Yoshimori, A. Uchida, K. Yoshimura, and P. Davis, “Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback,” Opt. Express 17, 10025–10034 ( 2009). [CrossRef] [PubMed]

]. Conversely, the optical frequency detuning can play a major role in the synchronization [18

P. Rees, P. S. Spencer, I. Pierce, S. Sivaprakasam, and K. A. Shore “Anticipated chaos in a non symmetric coupled external-cavity-laser system” Source: Phys. Rev. A 68, 033818-1–033818-7 ( 2003).

, 7

J. M. Buldú, T. Heil, I. Fischer, M. C. Torrent, and J. Garcia-Ojalvo, “Episodic Synchronization via Dynamic Injection,” Phys. Rev. Lett. 96, 024102-1–024102-4 ( 2006) [CrossRef]

, 10

I. Oowada, H. Ariizumi, M. Li, S. Yoshimori, A. Uchida, K. Yoshimura, and P. Davis, “Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback,” Opt. Express 17, 10025–10034 ( 2009). [CrossRef] [PubMed]

].

With a fourth order Runge-Kutta algorithm numerical data series were obtained for Ei (t) and these gave the normalized intensity series |Ei (t)|2. The time scales in the integrations and the equations parameters, if not stated otherwise, were fixed as: dt=1 ps, 1/τp =282 ns-1, N 0=1.5×108, ε=5×10-7, κ=γ=22 ns-1, 1/τs =1.66 ns-1. The times, τF and τC , are in the nanosecond range.

Calculations with zero frequency detuning and equal parameters and pump currents (Ji =1.013), corresponding to symmetrical systems, produced numerical time series with leadership exchange, again in agreement with the experiment. Segments of these data series are shown in figure 3, to be compared with the experimental segments in figure 2. Calculation with one laser having sufficiently higher pump current gives synchronized LFF with the laser of higher pump current always leading in time, as observed in the experiments. More on the sensitivity of the calculated dynamics with respect to changes on the parameters of equations 3, will be discussed below. Let us emphasize that no stochastic terms are used in the equations. The apparent random time distribution of the LFF events in each laser as well as the time leadership switching in the synchronism is due to the excitable nature of LFF as discussed in [14

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E 55, 6414–6418 ( 1997). [CrossRef]

, 16

J. F. Martinez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, “Coarse Grained Variables and Deterministic Chaos in an Excitable System,” Phys. Rev. Lett. 100, 044101-1–044101-4 ( 2008). [CrossRef]

]. If this excitable nature of LFF occurs for single lasers with feedback it also will be manifest as the fast fluctuations of one laser excites the LFF of the other, in a coupled scheme. It must be observed that the current argument do not eliminate the controversial possibility of the main origin of LFF in single lasers as due to external noise or internal fast fluctuations.

Fig. 3. Typical segment of the numerical time series of (a) laser 1 and (b) laser 2 LFF power drops under synchronized condition. Notice that the first drops are isochronous while time leadership was interchanged between drops two and three. τF =10 ns and τC =10.05 ns.

Numerical time series with ~104 pairs of drops were coarse grain filtered [16

J. F. Martinez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, “Coarse Grained Variables and Deterministic Chaos in an Excitable System,” Phys. Rev. Lett. 100, 044101-1–044101-4 ( 2008). [CrossRef]

] with a 1 ns time constant, and used to extract histograms of time delays between the drops. The switching of leadership from symmetrical systems was also examined and no correlation was found between consecutive values of ΔTi , up to second order conditional probability. This is indication of a Markov process, obtained despite the fact that the data came from deterministic numerical equations with recurrence times τF and τC . Such result is consistent with an interpretation of a pair of excitable systems created by high dimensional chaos with largely different time scales. The ultrafast pulses (scale of hundreds of picoseconds) existing in each laser acts as excitation pulses to trigger the LFF drops. The time difference (T i+1-Ti ≫τF ,τC ) among the LFF drops of each of the two lasers appear as stochastic withoutmemory [16

J. F. Martinez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, “Coarse Grained Variables and Deterministic Chaos in an Excitable System,” Phys. Rev. Lett. 100, 044101-1–044101-4 ( 2008). [CrossRef]

]. The experimental data show the same lack of correlation for the delay times of LFF synchronism in symmetrical conditions. The histograms associated to the data series of figures 2 and 3, are shown in figure 4.

With τF =10 ns and τC =10 ns, equation 1 with mi =±1 and ni =±1, predicts ΔTi =0,±10ns and ±20ns. Indeed, the histograms show that isochrony occurs, along with time leadership exchange events. The major probabilities are for events with ΔTi =0 and ±10 ns, with few events at ±20 ns and some in ±30 ns on the experimental data. It is important to mention that for the calculations with totally symmetrical lasers, (figure 4 (b)), τC ≠τF (τC =10.05 ns) a small difference (τF =10 ns and τC =10.05 ns) was necessary to give the non isochronous events. In fact such differences which are intrinsically present in the experiments change dramatically the amplitude of the peaks in the histograms but haveminor effect on the values of the allowed delays in synchronized LFF. The robustness of the relation between delay times and the condition of LFF synchronization according to equation 1 was also inspected for small, up to 10%, of relative variations of τF and τC . The sensitivity of the amplitudes of the histograms with some of the lasers parameters is very stong. Within the equal optical frequency calculations we could observe the dependence on the lasers currents, as given in figures 4 (b) and (c). As the pump currents were made different, the laser with higher current begins to dominate, presenting earlier LFF drops.

Fig. 4. Histograms of the time delays between LFF pulses of the two synchronized lasers with τF =10 ns and τC =10 ns(a) experimental, (b) numerical calculated with no optical detuning and equal parameters and (c) numerical with laser 2 and 1 having currents 1.014 and 1.013 above threshold, respectively.

Another typical experimental and numerical set of histograms, where mi and ni change within the same synchronized evolution, is shown in figure 5. This is a case of symmetrical system where the laser parameters are equal but the values for time of coupling and feedback, were chosen to give unstable isochrony: τC =15 ns and τF =10 ns. Accordingly, in equation 1 these values prevent ΔTi =0, as seen by direct substitution of small integers(mi =±1). Thus, in a large time scale synchronism of the LFFs holds, but there is always a finite time lead between the two lasers. The leadership can be exchange but events of simultaneous drops are excluded. A special case of this condition is the original paper by Heil it et al. [19

T. Heil, I. Fischer, W. Elssser, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86, 795–798 ( 2001). [CrossRef] [PubMed]

] describing non isochrony in the synchronized LFF of two diode lasers without feedback and optically coupled.

Fig. 5. Histograms from experimental (a) and numerical (b) series for the time delays between LFF drops of the two synchronized lasers: τF =10 ns and τC =15 ns. These delay values do not allow isochrone events, according to Eq. 1

When τC ≫τF we verified numerically that equation 1 still holds. The value of mi is always ±1 while ni assumes a large range of values. Isochrony is absent and the dominant events occur with ni =0, corresponding to delays of ±τC . Nonsymmetrical systems, as mentioned above, also follow equation 1. A calculation with the two lasers having the same current Ji =1.013 is given in figure 6 (a) while figure 6 (b) shows what happens when laser 1 has its current augmented to J 1=1.025. The laser with higher current is always the time leader, even though sometimes its leading time jumps between τC and τC -τF . These events correspond to LFF drops with mi =1 and ni =0 changing to mi =1 and ni =+1 along the dynamical evolution with synchronized LFF excitation spikes.

Fig. 6. Histograms of the numerical delay times between synchronized LFF drops of the lasers with τF =10 ns and τC =15 ns and showing the effect of asymmetric conditions in the pump currents. In (a). Ji =1.013 for the two lasers and in (b) J 1=1.025 and J 2=1.013, making Laser 1 the time leader.

Optical frequency detuning between the lasers was investigated in numerical solutions. The results for coupled lasers as in Eq. 3 verifies the previous results established in master-slave diode lasers synchronization [18

P. Rees, P. S. Spencer, I. Pierce, S. Sivaprakasam, and K. A. Shore “Anticipated chaos in a non symmetric coupled external-cavity-laser system” Source: Phys. Rev. A 68, 033818-1–033818-7 ( 2003).

, 19

T. Heil, I. Fischer, W. Elssser, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86, 795–798 ( 2001). [CrossRef] [PubMed]

, 7

J. M. Buldú, T. Heil, I. Fischer, M. C. Torrent, and J. Garcia-Ojalvo, “Episodic Synchronization via Dynamic Injection,” Phys. Rev. Lett. 96, 024102-1–024102-4 ( 2006) [CrossRef]

]: With other parameters equal, the laser with higher optical frequency is advanced in time. Our new results, for bidirectional coupling also shows the higher optical frequency laser leading the synchronized LFF drops, even when there is intermittent fluctuation in the advanced time difference. This is shown in figure 7. A striking difference is observed in the amplitudes of the histograms for Ω/2π=+1 GHz and Ω/2π=-1 GHz, even though the ΔTi delays remain at the same positions in the two cases. The two sets of histograms are calculated from time series that are not symmetrical in their fast fluctuations as we changed the signal of Ω. This is the reason for the histograms not to be perfectely mirrored with respect to zero detuning (the peak at -5 ns is not reproduced with the same amplitude at +5 ns).

The physical origin of the intermittent interchange of delays in the synchronized lasers of our experiments may be attributed to the excitable nature of LFF in single diode lasers with feedback. According to Giudici et al. [14

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E 55, 6414–6418 ( 1997). [CrossRef]

], the LFF power drops in a diode laser with optical feedback have the features characteristic of an excitable system driven by external noise. The LFF spikes can also be interpreted as the excited spike induced by the fast fluctuations (hundred picoseconds) contained in the deterministic dynamics of these lasers [15

J. F. Martinez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, “Experimental Deterministic Coherence Resonance,”Phys. Rev. Lett. 93, 144101-1–144101-4 ( 2004). [CrossRef]

, 16

J. F. Martinez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, “Coarse Grained Variables and Deterministic Chaos in an Excitable System,” Phys. Rev. Lett. 100, 044101-1–044101-4 ( 2008). [CrossRef]

]. The case of two lasers synchronization treated as a pair of of excitable systems using a common external noise source has also been studied by Cisak it et al. [27

M. Cisak, F. Marino, R. Toral, and S. Balle, “Mechanism of Anticipating Synchronization in Excitable Systems,” Phys. Rev. Lett. 93, 114102-1–114102-4 ( 2004).

]. Considering the fast fluctuations of each laser as the equivalent noise that triggers their LFFs, the optical coupling makes each laser be subject to both fluctuations and so their LFFs will be excited by a correlated source of equivalent noise. Such fluctuations are correlated with peaks that depend on the feedback times and the inter-units delay time. Thus synchronized LFF drops can occur excited by fluctuantions which give an apparent random distribution to their delay times; these delay times allways happening at the peaks of the fast correlation.

Fig. 7. Histograms of the numerical delay times between synchronized LFF drops of the lasers with τF =10 ns and τC =15 ns and Ji =1.013, showing the effect of optical frequency detuning. Above, with Ω/2π=-1 GHz, laser 1 had higher frequency and was time leader in the LFF synchronized drops. Below, the optical frequency was higher for laser 2, which became time leader. See text for explanation on the difference between the two histograms

4. Conclusion

To summarize, from experiments with coupled diode lasers with feedback, corroborated by the numerical solutions of the corresponding rate equations, we discovered that the delay time in the synchronization of coupled excitable systems with feedback is controlled by a simple relation between feedback time and inter-coupling time. The dynamicsmay have one of the systems with a fixed time leadership or have its leading time switching values by discrete steps depending on the values of the coupling and feedback times. Nearly symmetrical systems can also have intermittent time leadership exchange always maintaining the excitable spikes synchronized in the large time scale. A simple equation that specify the intercombinations of feedback and coupling time to give the allowed values of delay in the synchronized dynamics was introduced. Coupled asymmetrical systems also follow the conditions for the allowed delay time between events. The various parameters of the systems strongly influence the probability of specific delays, distorting the histogram amplitudes, but, within the same range of variation, have no effect on the values of the allowed time delays. Our results extend the previous [19

T. Heil, I. Fischer, W. Elssser, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86, 795–798 ( 2001). [CrossRef] [PubMed]

] determination of the symmetry breaking and instability of expected isochrony in the synchronism of identical coupled chaotic systems without feedback. It also adds to the recently studied properties of isochronal synchronismpresented in [25

E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E 73, 066214-1–066214-4 ( 2006). [CrossRef]

] and [9

I. Kanter, N. Gross, E. Klein, E. Kopelowitz, P. Yoskovits, L. Khaykovich, W. Kinzel, and M Rosenbluh, “Synchronization of mutually coupled chaotic lasers in the presence of a shutter,” Phys. Rev. Lett. 98, 154101-1–154101-4 ( 2007). [CrossRef]

], who emphasize the potential application of these properties for encrypted communication. Effects of pump current and optical frequency detuning have been shown to follow observations reported for coupled diode lasers when at least one do not have optical feedback [18

P. Rees, P. S. Spencer, I. Pierce, S. Sivaprakasam, and K. A. Shore “Anticipated chaos in a non symmetric coupled external-cavity-laser system” Source: Phys. Rev. A 68, 033818-1–033818-7 ( 2003).

, 19

T. Heil, I. Fischer, W. Elssser, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86, 795–798 ( 2001). [CrossRef] [PubMed]

]. It must be cautioned, however, that a throughly exploitation of the many parameter space of the two lasers, which is beyond this work, is lacking to characterize the range of validity of the observations near symmetrical lasers operation. A formal mathematical treatment of the stable and unstable fast dynamics chaotic synchronization with delays having integer combinations between feedback and coupling time will be presented elsewhere. The phenomenon of selection condition in time delays is bound to appear generically in mutually coupled dynamical systems and to have applications in schemes of encrypted communications.

Acknowledgments

The authors acknowledge useful discussions with H. L. D. de S. Cavalcante, L. Gil, L. Columbo and J. Tredicce. Work partially supported by Brazilian Agencies Conselho Nacional de Pesquisa e Desenvolvimento (CNPq) and Fundação de Ciência de Pernambuco (FACEPE) and by a Brazil-France Capes-Cofecub project 456/04.

References and links

1.

A. Pikovsky, M. Rosenblum, and J. Kurths, “Synchronization: A Universal Concept in Nonlinear Sciences,” (Cambridge Nonlinear Science Series, 2003)

2.

R. Vicente, L. L Gollo, C. R. Mirasso, I. Fisher, and Gordon Pipa, “Dynamical relaying can yield zero time lag neuronal synchrony despite long conduction delays,” PNAS- Proceedings of the National Academy of Sciences of the USA 105, 17157–17162 ( 2008). [CrossRef]

3.

T. Sugawara, M. Tachikawa, T. Tsukamoto, and T. Y. Shimizu, “Observation of Synchronization in Laser Chaos,” Phys. Rev. Lett. 72, 3502–3505 ( 1994). [CrossRef] [PubMed]

4.

R. Roy and K. S. Thornburg, “Experimental Synchronization of Chaotic Lasers,” Phys. Rev. Lett. 72, 2009–2012 ( 1994). [CrossRef] [PubMed]

5.

Y. Liu, P.C. de Oliveira, M.B. Danailov, and J. R. Rios Leite, “Chaotic and periodic passive Q switching in coupled CO2 lasers with a saturable absorber,” Phys. Rev. A 50, 3464–3470 ( 1994). [CrossRef] [PubMed]

6.

R. Meucci, F. Salvadori, M. V. Ivanchenko, K. Al Naimee, C. Zhou, F. T. Arecchi, S. Boccaletti, and J. Kurths, “Synchronization of spontaneous bursting in a CO2 laser,” Phys. Rev. E 74, 066207-1 0662070-6 ( 2006). [CrossRef]

7.

J. M. Buldú, T. Heil, I. Fischer, M. C. Torrent, and J. Garcia-Ojalvo, “Episodic Synchronization via Dynamic Injection,” Phys. Rev. Lett. 96, 024102-1–024102-4 ( 2006) [CrossRef]

8.

X. Li, W. Pan, D. Ma, and B. Luo, “Chaos synchronization of unidirectionally injected vertical-cavity surfaceemitting lasers with global and mode-selective coupling,” Opt. Express 14, 3138–3151 ( 2006). [CrossRef] [PubMed]

9.

I. Kanter, N. Gross, E. Klein, E. Kopelowitz, P. Yoskovits, L. Khaykovich, W. Kinzel, and M Rosenbluh, “Synchronization of mutually coupled chaotic lasers in the presence of a shutter,” Phys. Rev. Lett. 98, 154101-1–154101-4 ( 2007). [CrossRef]

10.

I. Oowada, H. Ariizumi, M. Li, S. Yoshimori, A. Uchida, K. Yoshimura, and P. Davis, “Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback,” Opt. Express 17, 10025–10034 ( 2009). [CrossRef] [PubMed]

11.

Y. Takiguchi, H. Fujino, and J. Ohtsubo, “Experimental synchronization of chaotic oscillations in externally injected semiconductor lasers in a low-frequency fluctuations regime,” Opt. Lett. 24, 1570–1572 ( 1999). [CrossRef]

12.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438, 343–346 ( 2005). [CrossRef] [PubMed]

13.

G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quant. Semiclass. Opt. 7, 87–143 ( 1995). [CrossRef]

14.

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E 55, 6414–6418 ( 1997). [CrossRef]

15.

J. F. Martinez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, “Experimental Deterministic Coherence Resonance,”Phys. Rev. Lett. 93, 144101-1–144101-4 ( 2004). [CrossRef]

16.

J. F. Martinez Avila, H. L. D. de S. Cavalcante, and J. R. Rios Leite, “Coarse Grained Variables and Deterministic Chaos in an Excitable System,” Phys. Rev. Lett. 100, 044101-1–044101-4 ( 2008). [CrossRef]

17.

C. Masoller, “Anticipation in the Synchronization of Semiconductor Lasers with Optical Feedback,” Phys. Rev. Lett. 86, 2782–2785 ( 2001). [CrossRef] [PubMed]

18.

P. Rees, P. S. Spencer, I. Pierce, S. Sivaprakasam, and K. A. Shore “Anticipated chaos in a non symmetric coupled external-cavity-laser system” Source: Phys. Rev. A 68, 033818-1–033818-7 ( 2003).

19.

T. Heil, I. Fischer, W. Elssser, J. Mulet, and C. R. Mirasso, “Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers,” Phys. Rev. Lett. 86, 795–798 ( 2001). [CrossRef] [PubMed]

20.

J. F. M. Avila, R. Vicente, J. R. Rios Leite, and C. R. Mirasso “Synchronization properties of bidirectionally coupled semiconductor lasers under asymmetric operating conditions”, Phys. Rev. E 75, 066202-1–066202-6 ( 2007). [CrossRef]

21.

J. Mulet, C. R. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B: Quant. Semiclass. Opt. 6, 97–105 ( 2004). [CrossRef]

22.

E. A. Rogers-Dakin, J. García-Ojalvo, D. J. DeShazer, and R. Roy, “Synchronization and symmetry breaking in mutually coupled fiber lasers,” Phys. Rev. E 73, 045201 ( 2006). [CrossRef]

23.

I. Fischer, R. Vicente, J. M. Buldú, M. Peil, and C. R. Mirasso, “Zero-Lag Long-Range Synchronization via Dynamical Relaying,” Phys. Rev. Lett. 97, 123902-1–123902-4 ( 2006). [CrossRef]

24.

I. B. Schwartz and L. B. Shaw, “Isochronal synchronization of delay-coupled systems,” Phys. Rev. E 75, 046207-1–046207-5 ( 2007). [CrossRef]

25.

E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E 73, 066214-1–066214-4 ( 2006). [CrossRef]

26.

N. Gross, W. Kinzel, I. Kanter, M. Rosenbluh, and L. Khaykovich, “Synchronization of mutually versus unidirectionally coupled chaotic semiconductor lasers,” Opt. Commun. 264, 464–468 ( 2006). [CrossRef]

27.

M. Cisak, F. Marino, R. Toral, and S. Balle, “Mechanism of Anticipating Synchronization in Excitable Systems,” Phys. Rev. Lett. 93, 114102-1–114102-4 ( 2004).

28.

R. Lang and K. Kobayashi, “External Optical Feedback Effects on Semiconductor Injection Laser Properties,” IEEE J. Quantum Electron. QE-16, 347–355 ( 1980). [CrossRef]

OCIS Codes
(060.4510) Fiber optics and optical communications : Optical communications
(140.1540) Lasers and laser optics : Chaos
(140.5960) Lasers and laser optics : Semiconductor lasers
(190.3100) Nonlinear optics : Instabilities and chaos

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: July 29, 2009
Revised Manuscript: November 4, 2009
Manuscript Accepted: November 4, 2009
Published: November 10, 2009

Citation
Jhon F. Martinez Avila and J. R. Rios Leite, "Time delays in the synchronization of chaotic coupled lasers with feedback," Opt. Express 17, 21442-21451 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21442


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences," (Cambridge Nonlinear Science Series, 2003)
  2. R. Vicente, L. L Gollo, C. R. Mirasso, I. Fisher and Gordon Pipa, "Dynamical relaying can yield zero time lag neuronal synchrony despite long conduction delays," PNAS- Proceedings of the National Academy of Sciences of the USA 105, 17157-17162 (2008). [CrossRef]
  3. T. Sugawara, M. Tachikawa, T. Tsukamoto and T. Y. Shimizu, "Observation of Synchronization in Laser Chaos," Phys. Rev. Lett. 72, 3502-3505 (1994). [CrossRef] [PubMed]
  4. R. Roy and K. S. Thornburg, "Experimental Synchronization of Chaotic Lasers," Phys. Rev. Lett. 72, 2009-2012 (1994). [CrossRef] [PubMed]
  5. Y. Liu, P.C. de Oliveira, M. B. Danailov and J. R. Rios Leite, "Chaotic and periodic passive Q switching in coupled CO2 lasers with a saturable absorber," Phys. Rev. A 50, 3464-3470 (1994). [CrossRef] [PubMed]
  6. R. Meucci, F. Salvadori, M. V. Ivanchenko, K. Al Naimee, C. Zhou, F. T. Arecchi, S. Boccaletti and J. Kurths, "Synchronization of spontaneous bursting in a CO2 laser," Phys. Rev. E 74, 066207-1 0662070-6 (2006). [CrossRef]
  7. J. M. Buldu, T. Heil, I. Fischer, M. C. Torrent and J. Garcia-Ojalvo, "Episodic Synchronization via Dynamic Injection," Phys. Rev. Lett.  96, 024102-1-024102-4 (2006) [CrossRef]
  8. X. Li, W. Pan, D. Ma and B. Luo, "Chaos synchronization of unidirectionally injected vertical-cavity surfaceemitting lasers with global and mode-selective coupling," Opt. Express 14, 3138-3151 (2006). [CrossRef] [PubMed]
  9. I. Kanter, N. Gross, E. Klein, E. Kopelowitz, P. Yoskovits, L. Khaykovich, W. Kinzel and M Rosenbluh, "Synchronization of mutually coupled chaotic lasers in the presence of a shutter," Phys. Rev. Lett. 98, 154101-1-154101-4 (2007). [CrossRef]
  10. I. Oowada, H. Ariizumi, M. Li, S. Yoshimori, A. Uchida, K. Yoshimura and P. Davis, "Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback," Opt. Express 17, 10025-10034 (2009). [CrossRef] [PubMed]
  11. Y. Takiguchi, H. Fujino and J. Ohtsubo, "Experimental synchronization of chaotic oscillations in externally injected semiconductor lasers in a low-frequency fluctuations regime," Opt. Lett. 24, 1570-1572 (1999). [CrossRef]
  12. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera and K. A. Shore, "Chaos-based communications at high bit rates using commercial fibre-optic links," Nature 438, 343-346 (2005). [CrossRef] [PubMed]
  13. G. H. M. van Tartwijk and D. Lenstra, "Semiconductor lasers with optical injection and feedback," Quant. Semiclass. Opt. 7, 87-143 (1995). [CrossRef]
  14. M. Giudici, C. Green, G. Giacomelli, U. Nespolo and J. R. Tredicce, "Andronov bifurcation and excitability in semiconductor lasers with optical feedback," Phys. Rev. E 55, 6414-6418 (1997). [CrossRef]
  15. J. F. Martinez Avila, H. L. D. de S. Cavalcante and J. R. Rios Leite, "Experimental Deterministic Coherence Resonance,"Phys. Rev. Lett. 93, 144101-1-144101-4 (2004). [CrossRef]
  16. J. F. Martinez Avila, H. L. D. de S. Cavalcante and J. R. Rios Leite, "Coarse Grained Variables and Deterministic Chaos in an Excitable System," Phys. Rev. Lett. 100, 044101-1-044101-4 (2008). [CrossRef]
  17. C. Masoller, "Anticipation in the Synchronization of Semiconductor Lasers with Optical Feedback," Phys. Rev. Lett. 86, 2782-2785 (2001). [CrossRef] [PubMed]
  18. P. Rees, P. S. Spencer, I. Pierce, S. Sivaprakasam and K. A. Shore "Anticipated chaos in a non symmetric coupled external-cavity-laser system" Source: Phys. Rev. A 68, 033818-1- 033818-7 (2003).
  19. T. Heil, I. Fischer, W. Elssser, J. Mulet and C. R. Mirasso, "Chaos Synchronization and Spontaneous Symmetry-Breaking in Symmetrically Delay-Coupled Semiconductor Lasers," Phys. Rev. Lett. 86, 795-798 (2001). [CrossRef] [PubMed]
  20. J. F. M. Avila, R. Vicente, J. R. Rios Leite and C. R. Mirasso "Synchronization properties of bidirectionally coupled semiconductor lasers under asymmetric operating conditions", Phys. Rev. E 75, 066202-1- 066202-6 (2007). [CrossRef]
  21. J. Mulet and C. R. Mirasso and T. Heil and I. Fischer, "Synchronization scenario of two distant mutually coupled semiconductor lasers," J. Opt. B: Quant. Semiclass. Opt. 6, 97-105 (2004). [CrossRef]
  22. E. A. Rogers-Dakin, J. Garcıa-Ojalvo, D. J. DeShazer and R. Roy, "Synchronization and symmetry breaking in mutually coupled fiber lasers," Phys. Rev. E 73, 045201 (2006). [CrossRef]
  23. I. Fischer, R. Vicente, J. M. Buldu, M. Peil and C. R. Mirasso, "Zero-Lag Long-Range Synchronization via Dynamical Relaying," Phys. Rev. Lett. 97, 123902-1-123902-4 (2006). [CrossRef]
  24. I. B. Schwartz and L. B. Shaw, "Isochronal synchronization of delay-coupled systems," Phys. Rev. E 75, 046207-1-046207-5 (2007). [CrossRef]
  25. E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich and I. Kanter, "Stable isochronal synchronization of mutually coupled chaotic lasers," Phys. Rev. E 73, 066214-1-066214-4 (2006). [CrossRef]
  26. N. Gross, W. Kinzel, I. Kanter, M. Rosenbluh and L. Khaykovich, "Synchronization of mutually versus unidirectionally coupled chaotic semiconductor lasers," Opt. Commun. 264, 464-468 (2006). [CrossRef]
  27. M. Cisak, F. Marino, R. Toral and S. Balle, "Mechanism of Anticipating Synchronization in Excitable Systems," Phys. Rev. Lett. 93, 114102-1-114102-4 (2004).
  28. R. Lang and K. Kobayashi, "External Optical Feedback Effects on Semiconductor Injection Laser Properties," IEEE J. Quantum Electron. QE-16, 347-355 (1980). [CrossRef]

Cited By

 Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited